Bayesian inference for functional extreme events defined via partially unobserved processes

TL;DR

This paper develops a Bayesian two-step MCMC algorithm to infer the extremal behavior of spatial stochastic processes from scattered observations, enabling the estimation of $r$-Pareto processes for extreme event analysis.

Contribution

It introduces a novel Bayesian framework with a two-step MCMC approach to handle partially observed processes for extreme value inference, improving practical applicability.

Findings

The probability of classifying an observation as an $r$-exceedance converges to the true probability.

The Markov chain in the second step converges to the posterior distribution under certain assumptions.

Simulation studies demonstrate the method's effectiveness compared to standard Bayesian procedures.

Abstract

In order to describe the extremal behaviour of some stochastic process $X$, approaches from univariate extreme value theory are typically generalized to the spatial domain. In particular, generalized peaks-over-threshold approaches allow for the consideration of single extreme events. These can be flexibly defined as exceedances of a risk functional $r$, such as a spatial average, applied to $X$. Inference for the resulting limit process, the so-called $r$-Pareto process, requires the evaluation of $r(X)$ and thus the knowledge of the whole process $X$. In many practical applications, however, observations of $X$ are only available at scattered sites. To overcome this issue, we propose a two-step MCMC-algorithm in a Bayesian framework. In a first step, we sample from $X$ conditionally on the observations in order to evaluate which observations lead to $r$-exceedances. In a second step, we use these exceedances to sample from the posterior distribution of the parameters of the limiting $r$-Pareto process. Alternating these steps results in a full Bayesian model for the extremes of $X$. We show that, under appropriate assumptions, the probability of classifying an observation as $r$-exceedance in the first step converges to the desired probability. Furthermore, given the first step, the distribution of the Markov chain constructed in the second step converges to the posterior distribution of interest. The procedure is compared to the Bayesian version of the standard procedure in a simulation study.